score:5
Yes. From the site you link to (which is also explained here)
it is quite clear that late night check in is available from 18:00 - 20:00 the evening before if your flight is scheduled to leave before 11 a.m. the next day which is the case for you.
I tweeted Lufthansa and they were quick to confirm the same info. Also Stuttgart Airport confirmed on Twitter, pointing to this part of their website (in German).
Regarding safety I have no personal experience but I would trust them as much as checking in your luggage the morning of your flight. Your luggage will be stored in the airport over night and unless you have anything really valuable in there I'd err on the side of trust here. I assume you will already get your boarding pass(es) when checking your luggage but for sure your luggage tags.
Finally about the chocolate, I see no reason why you could not take it as a carry-on but likely it is colder in the hold and less hassle for you to carry around.
Upvote:4
I have only used the Lufthansa late night check-in at Frankfurt, but as I expect procedures in Stuttgart to be the same, I'll post my experiences as an answer:
You get both your boarding passes and luggage tags when you drop your bag in the evening, so yes, the next day you go straight to security. Note that you can also combine the late night check in with online and mobile check-in, where you print the boarding pass at home or have it on your phone.
Maybe the general hint not to put valuables into your hold luggage is a little more critical here, as it will be sitting around the airport for some hours and (probably) be handled by a few employees more than when checking in in the morning, giving a malicious employee a better chance to find it before it gets loaded. But in general I back @mts, safety is essentially the same no matter when you check in.
If you need some final reassurance: I used the late night check-in two or three times without any issues, and I wouldn't hesitate to do it again (I was actually kind of sad when I saw that KLM doesn't offer this service at STR, as I will be flying with them next).